We give a characterization of the effect of sequences of pivot operations on a graph by relating it to determinants of adjacency matrices. This allows us to deduce that two sequences of pivot operations are equivalent iff they contain the same set S of vertices (modulo two). Moreover, given a set of vertices S, we characterize whether or not such a sequence using precisely the vertices of S exists. We also relate pivots to perfect matchings to obtain a graph-theoretical characterization. Finally, we consider graphs with self-loops to carry over the results to sequences containing both pivots and local complementation operations.
We give a combinatorial interpretation of the determinant of a matrix as a generating function over Brauer diagrams in two different but related ways. The sign of a permutation associated to its number of inversions in the Leibniz formula for the determinant is replaced by the number of crossings in the Brauer diagram. This interpretation naturally explains why the determinant of an even antisymmetric matrix is the square of a Pfaffian.
We prove that the maximum determinant of an $n \times n $ matrix, with entries in $\{0,1\}$ and at most $n+k$ non-zero entries, is at most $2^{k/3}$, which is best possible when $k$ is a multiple of 3. This result solves a conjecture of Bruhn and Rautenbach. We also obtain an upper bound on the number of perfect matchings in $C_4$-free bipartite graphs based on the number of edges, which, in the sparse case, improves on the classical Bregman's inequality for permanents. This bound is tight, as equality is achieved by the graph formed by vertex disjoint union of 6-vertex cycles.
The determinant method of Kasteleyn gives a method of computing the number of perfect matchings of a planar bipartite graph. In addition, results of Bernardi exhibit a bijection between spanning trees of a planar bipartite graph and elements of its Jacobian. In this paper, we explore an analogue of Bernardi's results, providing a canonical simply transitive group action of the Kasteleyn cokernel of a planar bipartite graph on its set of perfect matchings, when the planar bipartite graph in question is of the form $G^+$, as defined by Kenyon, Propp and Wilson.
We consider the computational problem of finding short paths in the skeleton of the perfect matching polytope of a bipartite graph. We prove that unless $P=NP$, there is no polynomial-time algorithm that computes a path of constant length between two vertices at distance two of the perfect matching polytope of a bipartite graph. Conditioned on $P\neq NP$, this disproves a conjecture by Ito, Kakimura, Kamiyama, Kobayashi and Okamoto [SIAM Journal on Discrete Mathematics, 36(2), pp. 1102-1123 (2022)]. Assuming the Exponential Time Hypothesis we prove the stronger result that there exists no poly…
This paper presents an algorithm to construct a weighted adjacency matrix of a plane bipartite graph obtained from a pretzel knot diagram. The determinant of this matrix after evaluation is shown to be the Jones polynomial of the pretzel knot by way of perfect matchings (or dimers) of this graph. The weights are Tutte's activity letters that arise because the Jones polynomial is a specialization of the signed version of the Tutte polynomial. The relationship is formalized between the familiar spanning tree setting for the Tait graph and the perfect matchings of the plane bipartite graph above.…
An exploration of the permanent-determinant method
永久行列式方法及其推广的Hafnian-Pfnian方法是由P.W.提出的计算平面图完美匹配的方法。卡斯泰琳我们提出了几个新的技术和参数有关的永久行列式的后果枚举组合。以下是这些技术的一些结果：1。若球面上4 n个顶点的二部图在对映映射下不变，则匹配数是商图匹配数的平方。2.度不超过3的图的边图的匹配数是2的幂。3.三个Carlitz矩阵的行列式计数a x b x c平面分区都有相同的上核。4.平面剖分的两个对称类几乎不需要计算就可以列举出来。
图的完美匹配的计数等价于在统计物理中有应用的二聚体问题。一个图$G$称为$n$ -旋转对称的，如果$n$阶的循环群是$G$的自同构群的子群. Jockusch（完美匹配和完满平方，J. Combin. Theory Ser.A，67（1994），100-115）和Kuperberg（An exploration of the permanent-determinant method，Electron.J.Combin.，5（1998），#46）独立证明了：如果$G$是一个具有$2n$ -旋转对称的$N$阶平面二部图，则$G$的完美匹配数可以表示为$N/2n$阶$n$行列式的乘积.在本文中，我们给这个结果一个新的介绍。我们利用这个结果计算了具有$2n$符号对称性的体平面二分格的熵。我们得到了两种类型的圆柱的完美匹配和熵的数量的显式表达式。利用Kenyon、Okounkov和Sheffield关于环面熵的结果，（Dimers and amoeulum，Ann.Math.163（2006），1019- 1056）以及Salinas和Nagle（层状氢键$SnCl^2\cdot 2H_2O$晶体中的相变理论，物理评论B，9（1974），4920- 4931），我们证明了每个考虑的圆柱和它相应的环面具有相同的熵。最后，我们提出了一些问题。
It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed versions of the latter three bijections for triangular regions. We first investigate the tileability of triangular regions by lozenges. Then we use perfect matchings and families of non-intersecting latti…
Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs
我们提出了一个确定性的方式分配小（日志位）的权重的二分平面图的边缘，使最小重量完美匹配成为唯一的。隔离引理（Mulmuley et al. 1987）中所描述的实现了相同的一般图使用随机加权方案，而我们可以做到这一点时，确定性限制到二分平面图。因此，我们减少了决策和建设版本的匹配问题，以测试是否矩阵是奇异的，根据承诺，其行列式是0或1，从而获得一个高度并行的SPL算法的二分平面图。这改进了先前已知的非均匀SPL的界限（Allender et al. 1999）和$NC^2$的界限（Miller and Naor 1995，Mahajan and Varadarajan 2000）。这也重新点燃了希望，获得一个确定性的并行算法，构造一个完美的匹配在非二部平面图，这已经开放了很长一段时间。我们的技术是基本和简单的。
We study the problem of sampling a uniformly random directed rooted spanning tree, also known as an arborescence, from a possibly weighted directed graph. Classically, this problem has long been known to be polynomial-time solvable; the exact number of arborescences can be computed by a determinant [Tut48], and sampling can be reduced to counting [JVV86, JS96]. However, the classic reduction from sampling to counting seems to be inherently sequential. This raises the question of designing efficient parallel algorithms for sampling. We show that sampling arborescences can be done in RNC. For se…
The classical 1961 solution to the problem of determining the number of perfect matchings (or dimer coverings) of a rectangular grid graph -- due independently to Kasteleyn and to Temperley and Fisher -- consists of changing the sign of some of the entries in the adjacency matrix so that the Pfaffian of the new matrix gives the number of perfect matchings, and then evaluating this Pfaffian. Another classical method is to use the Lindström-Gessel-Viennot theorem on non-intersecting lattice paths to express the number of perfect matchings as a determinant, and then evaluate this determinant. In …
MacMahon enumerated the plane partitions in an $a \times b \times c$ box. These are in bijection to lozenge tilings of a hexagon, to certain perfect matchings, and to families of non-intersecting lattice paths. In this work we consider more general regions, called triangular regions, and establish signed versions of the latter three bijections. Indeed, we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. A combinatorial argument involving a new method, called resolution of a puncture, then shows that the signs are in fact equivalent. …
